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Full text of "Treatise On Analysis Vol-Ii"

6 POSITIVE LINEAR FORMS, POSITIVE HILBERT FORMS 349 (ii) For all x, y in A we have (15.6.2.2) \f(y*x)\2 £/(***)/(/». (iii) If A has a unit element e, then (15.6.2.3) /(**) =7(7), (15.6.2.4) |/(x)|2 £/(«)/(***). To prove (i), express that g(x + y9x+y) = g(x, x) + g(x9 y) + #0>, x) + g(y, y) is real: by virtue of the hypothesis (15.6.1), we obtain changing x into ix, this becomes 3$(g(x9 y)) = @%(g(y, x)), hence g(y, x) = g(x, y). Assertion (ii) is the Cauchy-Schwarz inequality (6.2.1 ) applied to g. The relations (iii) are obtained by replacing y by e in (15.6.2.1) and (15.6.2.2), and using (15.4.2). The hermitian form g so obtained from / is not arbitrary, because it clearly satisfies the relation (15.6.3) g(xy,z)=g(y,x*z) for all x, y, z in A. We therefore make the following definition: A positive Hilbert form on the algebra with involution A is a positive hermitian form on A x A which satisfies the relation (15.6.3). If A has a unit element, every positive Hilbert form g comes as above from a positive linear form/: we have only to define /(jc) = g(x, e). But we shall see later that this is no longer the case when A has no unit element (15.7.4). We shall now show that there are remarkable relationships between Hilbert forms on A x A and representations of A. In the first place, every representation s^ U(s) of A in a Hilbert space H gives rise to positive linear forms (and hence to positive Hilbert forms) in the following way: for each XQ e H, if we put (15.6.4) /*(*)= («(s) • x0 | *0)f(y*x) is a positive hermitian form on